Given two matrices $X$ and $Y$, it's easy to take the inverse of their Kronecker product:
$(X\otimes Y)^{-1} = X^{-1}\otimes Y^{-1}$
Now, suppose we have some diagonal matrix $\Lambda$ (or more generally an easily inverted matrix, or one for which we already know the inverse). Is there a closed-form expression or efficient algorithm for computing $(\Lambda + (X\otimes Y))^{-1}$?
yes there is.
See equation 5 in http://books.nips.cc/papers/files/nips24/NIPS2011_0443.pdf
Stegle et al. Efficient inference in matrix-variate Gaussian models with iid observation noise